Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Entrance — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The wall's condition, on its way to the middle. Five profiles across the half-channel, from just inside the entrance to fully developed, each drawn at the station where it occurs. The march starts from a slab of uniform flow and never assumes a shape: what arrives at the far end is a parabola, with a centre-line speed of 1.4979 times the mean against the exact 3/2 and a momentum flux of 1.1995 against 6/5. Notice what the middle does while the edges are being slowed: it speeds up, because the flow rate is held, and that acceleration is what the entrance's extra pressure drop pays for.

    How far before a duct forgets what was fed into it

    A pipe is always drawn with its answer already in place. Getting there takes a distance proportional to the Reynolds number, which means a more viscous fluid is done sooner — and the entrance costs a fixed number of dynamic pressures however long the pipe is.

    part 1 · viscous
  2. How fast a duct forgets each part of its inlet. The decay rate along the duct for each mode of a disturbance to the developed profile. It goes as the square of the mode number, so the third mode is forgotten nine times faster than the first and the tenth a hundred times faster.

    A duct that forgets everything but one number

    Whatever is fed into a pipe, what survives a little way down it is one shape. The disturbance's higher modes decay as the square of their mode number, so the sixth is gone in thirteen centimetres where the first survives four and a half metres — and the entrance length is that one mode's decay rate and an arbitrary threshold.

    part 2 · viscous

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